Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=4 for quadruplets. If a gap is larger than all preceding gaps, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps. This sequence suggests that maximal gaps between prime quadruplets are O(log^5(p)). - Alexei Kourbatov, Jan 04 2012

REFERENCES

Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.

(1) Upper bound: gaps between prime quadruplets (p, p+2, p+6, p+8) are smaller than 0.241*(log p)^5, where p is the prime at the end of the gap.

(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a*(log(p/a)-0.55), where a = 0.241*(log p)^4 is the average gap between quadruplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.

Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.

Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.241 is reciprocal to the Hardy-Littlewood 4-tuple constant 4.15118... (End)

EXAMPLE

The first prime quadruplets are (5,7,11,13) and (11,13,17,19), so a(1)=11-5=6. The next quadruplet is (101,103,107,109), so a(2)=101-11=90. The following quadruplet is 191,193,197,199 so 90 remains the record and no terms are added.

CROSSREFS

A229907 lists initial primes in quadruplets preceding the maximal gaps. A113403 lists the corresponding primes at the end of the maximal gaps. Cf. A008407, A007530.